Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What is the antiderivative of $(2x+7)^{1/2}$?

My understanding is that it would be $\frac 23 \times(2x+7)^{3/2}$ but according to the source I am working from the answer is $\frac 13 \times (2x+7)^{3/2}$.

share|cite|improve this question
11  
Differentiate your answer and you will see why. HINT: Chain rule. – mickep Mar 2 at 10:00
    
You probably forgot to process the coefficient of $x$. – Yves Daoust Mar 2 at 10:13
    
Why do I need to process the coefficient of x before I take the antiderivative? – user252704 Mar 2 at 11:57
    
Because it is going to be processed when you take the derivative. If it changes in one direction, it has to change in both directions, or else the derivative of an antiderivative wouldn't equal the original function. – kevinsa5 Mar 2 at 14:13

$$\int\sqrt{(2x+7)}dx$$

Set $t=2x+7$ and $dt=2dx$

$$=\frac 1 2\int\sqrt tdt=\frac{t^{3/2}}{3}+\mathcal C=\frac{(2x+7)^{3/2}}{3}+\mathcal C$$

share|cite|improve this answer
    
Why multiply it by 1/2 and also by 2? – user252704 Mar 2 at 12:01
4  
@user252704 its because $dt=2dx$...therefore $dx=\dfrac{dt}{2}$...now all we need to do is to plug in the values... – manshu Mar 2 at 12:39

Notice:

$$\int x^n\space\text{d}x=\frac{x^{1+n}}{1+n}+\text{C}$$



$$\int\sqrt{2x+7}\space\text{d}x=$$


Substitute $u=2x+7$ and $\text{d}u=2\space\text{d}x$:


$$\frac{1}{2}\int\sqrt{u}\space\text{d}u=\frac{1}{2}\int u^{\frac{1}{2}}\space\text{d}u=\frac{1}{2}\cdot\frac{u^{1+\frac{1}{2}}}{1+\frac{1}{2}}+\text{C}=\frac{u^{\frac{3}{2}}}{3}+\text{C}=\frac{(2x+7)^{\frac{3}{2}}}{3}+\text{C}$$

share|cite|improve this answer
    
Why multiply it by 1/2 and also by 2? – user252704 Mar 2 at 12:01
1  
To make the substitution clear – Jan Eerland Mar 2 at 12:59

$$(2x+7)^{1/2}=2^{1/2}(x+\frac72)^{1/2}\to2^{1/2}\frac1{\frac32}(x+\frac72)^{3/2}=\frac{2^{3/2}}3(x+\frac72)^{3/2}=\frac{(2x+7)^{3/2}}3.$$

share|cite|improve this answer
    
Why do I need to process the coefficient of x before I take the antiderivative? – user252704 Mar 2 at 11:58
    
I noticed that you didn't handle it properly so this is a workaround. – Yves Daoust Mar 2 at 12:13
    
@user252704, it's like the chain rule in reverse... (2x+7)^(1/2) is like f(g), where f(x) = x^(1/2) and g(x) = 2x+7, except here you have to remember to process the ANTIderivative of g instead of the derivative – sig_seg_v Mar 2 at 16:50

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.